Entanglement islands and cutoff branes from path-integral optimization

Recently it was proposed that, the AdS/BCFT correspondence can be simulated by a holographic Weyl transformed CFT$_2$, where the cut-off brane plays the role of the Karch-Randall (KR) brane \cite{Basu:2022crn}. In this paper, we focus on the Weyl transformation that optimizes the path integral computation of the reduced density matrix for a single interval in a holographic CFT$_2$. When we take the limit that one of the endpoint of the interval goes to infinity (a half line), such a holographic Weyl transformed CFT$_2$ matches the AdS/BCFT configuration for a BCFT with one boundary. Without taking the limit, the induced cutoff brane becomes a circle passing through the two endpoints of the interval. We assume that the cutoff brane also plays the same role as the KR brane in AdS/BCFT, hence the path-integral-optimized purification for the interval is in the island phase. This explains the appearance of negative mutual information observed in \cite{Camargo:2022mme}. We check that, the entanglement entropy and the balanced partial entanglement entropy (BPE) calculated via the island formulas, exactly match with the RT formula and the entanglement wedge cross-section (EWCS), which are allowed to anchor on the cutoff brane.


Introduction
Recently, a new rule to compute the entanglement entropy in gravitational systems, namely the island formula [3][4][5][6][7], has led to a remarkable new perspective to understand the long standing puzzle of the black hole information paradox [8].The development of the island formula originates from the celebrated Ryu-Takayanagi (RT) formula [9,10], which relates the entanglement entropy in holographic CFT to the minimal surfaces in the dual AdS bulk.Then the RT formula was refined to the quantum extremal surface (QES) formula [11][12][13] with the quantum correction taken into account.In [3,4] the QES formula was used to compute the entanglement entropy for the Hawking radiation of an evaporating black hole, and the result is consistent with the Page curve.Remarkably, the QES formula suggests that, a region in black hole interior I, which we call the entanglement island, should be considered as a part of the Hawking radiation R.And the entanglement entropy for R is given by the island formula, which contains an optimization for all possible I. Soon, it was argued that when we apply the replica trick in gravitational theory in a path-integral representation, we should take into account the new geometric configurations with wormholes in the replica manifold.
When the replica wormhole configuration dominate the path-integral, the entanglement entropy should be computed by the island formula [6,7].Note that, replica wormhole arguments apply to generic gravitational theories and do not rely on the existence of holography.When a gravitational system is in island phase, it is interesting to note that for given a region R and its island region Is(R), the state of the island region Is(R) can be reconstructed from the information in the R region.The existence of such a reconstruction is implied by the island formula (1.1) following the same line of argument for entanglement wedge reconstruction [6,[14][15][16].And an explicit scheme of reconstruction in some simple models of quantum gravity was given in [6] based on the Petz map [17][18][19][20].In other words the state of Is(R) is encoded in the state of R, hence the degrees of freedom in Is(R) are not independent and the Hilbert space of the whole system becomes non-factorizable.In [1] this property was further refined as the so-called self-encoding property between spacelikeseparated sub-regions, and a possible generalization of this property to systems beyond gravitational theories was explored in [1].
The AdS/BCFT correspondence [21,22] is a commonly used context under which the entanglement islands emerges.It was found that the AdS/BCFT configuration can be perfectly simulated by models of Weyl transformed holographic CFT 2 [1,23,24].The particular Weyl transformations under consideration are special as they depend on the cutoff scale of theory, which effectively introduces finite cutoff scale to the region where entanglement islands can emerge.The Weyl transformations will induce a cutoff brane in the bulk.It was shown in [1,23,24] that, the cutoff brane plays the same role as a KR brane where the RT surfaces are allowed to anchor.If we adjust the Weyl transformation such that the cutoff brane overlaps with the KR brane in the AdS bulk, we can exactly reproduce the main features of the AdS/BCFT configuration.
In this paper, we will consider a special Weyl transformation that optimizes the path integral computation of the reduced density matrix of an interval in a holographic CFT 2 [25,26], and use it to simulate the AdS/BCFT configuration.This means the path-integraloptimized purification of the interval is in island phase.We give non-trivial consistency check for this simulation by computing the entanglement entropy and the BPE on the field theory side via the island formulas, and find that the results match with the RT surfaces and EWCSs that are allowed to anchor on the cutoff brane.
In section 2, we briefly introduce the setup of the AdS 3 /BCFT 2 correspondence and its simulation via a holographic Weyl transformed CFT 2 .In section 3, we introduce the computation of the reduced density matrix for an interval and the Weyl transformation that optimizes the path integral.Furthermore, we derive the cutoff branes for holographic CFT 2 under such Weyl transformations, which coincide with the KR branes in AdS/BCFT.In section 4, assuming that the path-integral-optimized purification for an interval is in island phase, we calculate the entanglement entropy for sub-intervals of the interval from both sides of the holography and find agreement.Also we show that the island phase perspective for this purification solves a puzzle of the existence of negative mutual information in this purification [2].We give a summary in section 5.In the Appendix, assuming the cutoff brane plays the role of the KR brane, we study the EWCS for bipartite sub-interval AB and the corresponding balanced partial entanglement entropy (BPE) in island phases, and test their correspondence.This gives further evidence for our claim that, the path-integraloptimized purification is in island phase.

AdS/BCFT and holographic Weyl transformed CFT
In the AdS/BCFT correspondence [21] (or equivalently the Karch-Randall braneworld [27,28]), the boundary theory is a d dimensional CFT d with boundaries, and the gravity dual is an AdS d+1 gravity in d + 1 dimensions bounded by a co-dimension one KR brane, where N denotes the bulk AdS spacetime, Q denotes the Karch-Randall (KR) brane anchored at the boundary of the CFT and T is the tension of Q.In the bulk, the von Neumann boundary conditions are imposed on the KR brane.In this paper, we focus on the d = 2 case and write the bulk metric in terms of the following coordinates, where ℓ is the AdS radius which we will set to be unit in the rest of this paper, and the two sets of coordinates are related by z = y cosh −1 ρ, x = −y tanh ρ . (2.3) x 0 For a BCFT with one boundary settled at x = 0, the corresponding KR brane locates at ρ = ρ 0 , where ρ 0 is a constant determined by the tension of the brane ρ 0 = arctanh T .On the other hand, let us consider the vacuum state of a holographic CFT 2 in flat background where ϵ represent the UV cutoff of the CFT.Then we perform a Weyl transformation characterized by a scalar field φ(x), hence the metric changes to be It can be understood that, the Weyl transformation changes the cutoff scale in a positiondependent way, ϵ → e −φ(x) ϵ . (2.6) The scalar φ(x) is chosen to be negative hence the Weyl transformation enlarges the cutoff scale.Accordingly, the entanglement entropy for a single interval A = [a, b] is modified to be [2,25,26] following the transformation rule of the two-point functions of twist operators under Weyl transformations.The above result is just the original entanglement entropy subjecting the absolute value of the scalar field at the endpoints.Holographically, the constant subjection of the entanglement entropy was understood as inserting cutoff spheres with radius |φ(x)| centered at the endpoints [1].In other words, when computing the entanglement entropy via the RT formula, we should exclude the portion of the RT surface inside the cutoff spheres, see Fig. 1.Interestingly, the cutoff sphere centered at (x, z) = (x 0 , ϵ) in AdS 3 is just a circle in flat background with radius α, [1,29], The particular Weyl transformation, that captures the main features of the AdS/BCFT with the KR brane settled at ρ = ρ 0 , is given by [ where κ is a constant.The cutoff brane [1] is defined as the common tangent line of all the cutoff spheres, which represent the boundary of the bulk cutoff region (see Fig. 2).In this case the cutoff brane is settled at which can be adjust to exactly overlap with the KR brane by setting To justify the application of the island formula in this Weyl transformed CFT, we need to give an additional assumption, which is that the x < 0 region was coupled to a gravity.The gravity is an AdS 2 gravity as the Weyl transformation changes the metric at the x < 0 region to AdS 2 .Later, we will return to this assumption in the discussion section.Then entanglement islands are allowed in this region according to the replica wormhole arguments [5,6].Furthermore, for simplicity we assume the AdS 2 gravity is a induced gravity as in [30], hence the area term in (1.1) will not appear.
On the field theory side, let us apply the island formula (1.1) to calculate S A for the interval A = [0, L] in this Weyl transformed CFT and assuming the island region Is(A) = [−a, 0), then we have (2.12) Plugging into (2.9), the above formula is minimized at a = L, hence This exactly matches the result in AdS/BCFT with ρ 0 = κ.On the gravity side, the holographic entanglement entropy is given by the area of the minimal extreme surface (the RT surface) which is homologous to A and is allowed to anchor at any of the cutoff spheres.As expected [1], the RT surface is just the circle emanating from the endpoint x = L on the boundary and anchored on the cutoff brane vertically, see the blue circle in Fig. 2. Provided κ = ρ 0 , this is exactly the RT surface anchored on the KR brane in the AdS/BCFT correspondence.
The above configuration exactly matches with the AdS/BCFT correspondence given ρ 0 = κ.The key for this simulation is that, we should adjust κ 1 such that the cutoff brane overlaps with the KR brane in the AdS/BCFT configuration which we simulate.Also, the coincidence between the two set-ups can be straightforwardly checked for the calculation of the entanglement entropy for any interval A = [a, b] in the x > 0 region, with phase transitions for the RT surfaces.

Weyl transformed CFT from path-integral optimization and AdS/BCFT
In the previous section, we simulated the AdS/BCFT configuration via the holographic Weyl transformed CFT 2 with the scalar field (2.9) and the constant κ adjusted to satisfy κ = ρ 0 .Note that, in this case the Weyl transformation is adjusted by hand such that the cutoff brane and the KR brane overlap.Since in the AdS/BCFT correspondence, the location of the KR brane is determined by details of the theory, including the tension and the boundary conditions of the brane.It is intriguing to ask what is special about the corresponding Weyl transformation (2.9) from the perspective on the field theory side.One of the main observation in this paper is that, the particular Weyl transformation (2.9) that makes the simulation a success is the one that optimizes the path integral computation for the reduced density matrix of the non-gravitational (or bath) region, which satisfies two key points: 1) it preserves the reduced density matrix ρ A for an interval A at a particular time, and 2) it minimizes the path-integral complexity C L [ϕ] defined later [25,31].
Let us start with the path integral computation for a quantum state of the whole system.Consider a 2d CFT on Euclidean flat space ds 2 = (dτ 2 + dξ 2 )/ϵ 2 = δ ab /ϵ 2 , where ϵ represents the UV cutoff.Under this metric the ground state |Ψ⟩ wave functional Ψ[ φ(ξ)] at τ = −ϵ is given by the Euclidean path integral on the half plane, Here the subscript of the wave function Ψ represents the metric of the Euclidean space where the path integral is performed.One can perform a Weyl transformation, which is a symmetry of the theory, to the metric, where the second equation is the boundary condition for the scalar field that characterizes the Weyl transformation.Then the state Ψ computed under the Weyl transformed metric is proportional to (3.1), where C L [ϕ] the Liouville action [32], This means the state Ψ is preserved under the Weyl transformation (3.2).In [25,31] It is easy to check that the above solution satisfies the boundary condition in (3.2).Note that, one can shift the above scalar field by a constant by choosing a different UV cutoff scale ϵ.
The above optimization procedure was generalized to optimizing the path integral computations that preserve the state (or the reduced density matrix) on a single interval A = [a, b], t = 0 as described in [26].This involves the Euclidean path integral over a complex plane (η, η) = (x + it, x − it) with the interval A cut open.Interestingly, one can relate this path integral to the one on a half plane by performing a conformal transformation which maps the interval A to an infinitely long line, Then we can optimize the Euclidean path integral on the half ω plane whose optimization gives (3.5).Finally, we may obtain the Weyl transformation by mapping (3.5) back to the (t, x) plane.Hence, we get the Weyl transformation that optimizes the path integral computation for the reduced density matrix ρ A of the interval A. More explicitly, the relation ϵ 2 τ 2 dωdω = e 2ϕ(x) dηdη gives us, where κ is a constant which depends on the choice of ϵ.Now we are ready to derive the cutoff brane induced by the above Weyl transformation, which is again the common tangent line of all the cutoff spheres.Firstly, according to (2.8) the cutoff spheres are circles on a time slice described by Then it is easy to check that the common tangent line of all the spheres is also a part of a circle which passes the two endpoints of the interval (see Fig. 3), cutoff brane: where z 0 = b−a 4 (e κ − e −κ ).Changing the constant κ will change the intersection angle between the cutoff brane and the asymptotic boundary.In Fig. 3, we give the cutoff brane for the same interval with different κ.
When we send one of the endpoint to infinity by taking the limit b → ∞, we are back to the BCFT with one boundary discussed in the previous section.Interestingly, under this limit the cutoff brane (3.9) becomes, z = (a − x) csch(κ) . (3.10) According to the coordinate transformation (2.3), the above equation is just ρ = κ if we put the other endpoint at the origin by setting a = 0.This exactly coincides with the cutoff brane (2.10), which is adjusted by hand to match the KR brane in the AdS/BCFT correspondence.Note that, if we choose an arbitrary smooth function ϕ(x) for the Weyl transformation, the shape of the cutoff brane could be arbitrary.It is remarkable that the Weyl transformations that optimize the path integral exactly reproduce the KR brane configurations in the AdS/BCFT correspondence for the BCFT with one boundary.Then it is intriguing to propose that, the cutoff brane configurations from path integral optimization can match to the KR branes in more generic AdS/BCFT setups.If this proposal is right, there should be KR brane configurations as circles (3.9) in the Poincaré AdS 3 bulk that homologous to the strip BCFT 2 with two boundaries.Since the cutoff brane is connected in the bulk, it belongs to the confined phase [21,22,33], rather than the deconfined phase where the brane is disconnected in the bulk 2 .Unfortunately, such configurations in AdS/BCFT were not confirmed in the literature yet.There are AdS/BCFT configurations where the KR brane on a time slice are circles, but the BCFT is not a strip.For example, in [22] the authors considered a BCFT on a round disk: τ 2 + x 2 ≤ L 2 in Euclidean spacetime, and find that the KR brane in AdS bulk dual is given by a sphere, where ρ 0 = arctanh T .For any fixed τ , the KR brane is also a portion of a circle.Also in [35], the KR branes connecting the two asymptotic boundaries in the eternal black hole are circles on a time slice when mapping to the Poincaré patch (see Fig. 6 in [35]).Similar configurations or KR branes were also proposed in [36].
4 The path-integral-optimized purification as a state in island phase In [37,38], it was claimed that the entanglement of purification (EoP) is the holographic dual of the EWCS in holographic theories.The optimization for the path integral that computes the reduced density matrix for an interval was first studied in [26] as a special purification for the interval, where they can compute the entanglement of purification (EoP), and confirm the claim of [37,38].In [26], this special purification is taken as a pure state in a normal quantum system without entanglement islands.However, it was pointed out in [2] that, there exist negative mutual information in this purification.On the other hand, following the [1,23,24], as well as the discussion in the previous section, we should understand the holographic CFTs under path-integral-optimized Weyl transformation as a simulation of the AdS/BCFT correspondence, hence we should take this purification as a pure state in island phase.In this section, we will compare between these two perspectives and show that, taking this path-integral-optimized purification as a state in island phase solves the puzzle of negative mutual information.
Let us consider a bipartite region in the boundary field theory with a partition AB ≡ A ∪ B, and ρ AB is the corresponding reduced density matrix.On a time slice, the entanglement wedge W AB of AB is the region enclosed by AB and the RT surface E AB .Given the entanglement wedge we can define the EWCS Σ AB as the minimal area cross-section of W AB separating the regions A and B. Let |Ψ⟩ ∈ H AA ′ ⊗ H BB ′ be any purification of ρ AB , the EoP between A and B is defined as [39] where we take the minimization over all possible purifications of AB and over all possible partitions of A ′ B ′ .The EoP is then given by the minimal value of S AA ′ .The authors in [26] start from the vacuum state of the holographic CFT 2 which duels to Poincaé AdS 3 and AB is an interval in this state, then they perform different Weyl transformations for the complement of AB (or A ′ B ′ ) to get a class of pure states computed by path integral since the Weyl transformations change the boundary condition for the path integral.As ρ AB is preserved by these Weyl transformations, the resulting pure states can be considered as different purification for ρ AB .The key observations of [26] are that, 1) the purification |Ψ⟩ where S AA ′ is minimized is exactly the one given by the Weyl transformation that optimizes the path-integral computation of ρ AB , i.e. the one characterized by (3.7).And 2), in the path-integral-optimized |Ψ⟩, the minimized S AA ′ over all the possible partition of the purifying system A ′ B ′ exactly matches the length of the EWCS, upon an additional choice of κ = 0.
More explicitly, let us consider the configuration shown in Fig. 4, where the purification of the mixed state ρ AB is path-integral optimized, hence the Weyl transformation applied on A ′ B ′ is characterized by (3.7) with κ = 0. Then we minimize the entanglement entropy3 by adjusting the partition point x = q and found that S AA ′ is minimized when hence, The authors of [26] studied the path-integral-optimized state |Ψ⟩ in no-island phase 4 .This indicates that the Hilbert space of the system factorizes as all the space-like separated degrees of freedom are independent, and the entanglement entropy for any interval [a, b] can be defined in a standard way and calculated following the formula (2.7).Nevertheless, it was pointed out in [2] that, under such a setup there exists negative mutual information in the path-integral-optimized state |Ψ⟩.Again, let us consider the path-integral-optimized purification of Fig. 4 with the partition point settled by (4.3), and assume that the purification is in the no-island phase.Then we can use the formula (2.7) to calculate the entanglement entropy for the following intervals, where, according to (3.7), we have ϕ(a) = ϕ(b) = 0. Also we can perform a straightforward calculation for the mutual information I(A ′ : B ′ ), where in the last line we used (4.3) and (3.7).As we can see, the mutual information I(A ′ : B ′ ) is negative, which indicates that the path-integral-optimized purification |Ψ⟩ is not a physical state.Following our previous discussion on the Weyl transformed CFT 2 , we turn to the perspective of taking the path-integral-optimized purification as a state in island phase.From this perspective, soon we will show that A ′ B ′ is the island region of AB, hence the state of A ′ B ′ is encoded in AB, which means the degrees of freedom in A ′ B ′ are not independent.This implies that the strong sub-additivity, which guarantees the positivity for mutual information, does not apply.So the new perspective straightforwardly solves the above puzzle of negative mutual information.Of course there are other possibilities 5  to avoid the negative mutual information, which we will not discuss further in this paper.
Let us give a more detailed comparison between the two perspectives on the pathintegral-optimized purification.For simplicity, we set κ = 0 hence the Weyl transformation matches with the one used in [26], and the cutoff brane (3.9) coincide with the RT surface of AB (see the left figure in Fig. 3).In the island perspective, the RT surfaces for subregions in AB are allowed to anchor on the cutoff brane, which results in a totally new understanding 5 One may consider other possibilities to avoid the negative mutual information by modifying the way we compute the entanglement entropy instead of using (2.7) and (3.7).For example, one may consider the physical degrees of freedom all locate on the boundary interval and the bulk brane, and consider the state/surface correspondence [40].In this context, A ′ and B ′ are sub-regions of the brane and the entanglement entropies are calculated by bulk geodesics connecting their endpoints, which exactly coincide with A ′ and B ′ in the case of Fig. 4.So we have S A ′ B ′ = S A ′ + S B ′ , and the mutual information we considered is zero.Nevertheless, the surface/state correspondence is not a context where we study the configurations with entanglement islands.Another possible solution is to keep using (4.3) and (3.7) but consider a constant shift for the scalar field, such that the mutual information (4.6) satisfies I(A ′ , B ′ ) ≥ 0. This equals to confining the constant term κ in (3.7) to satisfy κ ≥ log 2. Also the cutoff brane changes, see section A.3 for more discussion on the case of κ ̸ = 0.One shortcoming of this solution is that, the entanglement entropy S AA ′ on longer reproduces the EWCS for the entanglement wedge of WAB due to the shift of a constant, see the discussion section of [2] for details.Also it is not obvious that this confinement can guarantee the positivity of an arbitrary mutual information.on the entanglement structure of this purification and the corresponding geometric dual on the gravity side.
Firstly, let us consider the configuration depicted by the upper figure in Fig. 5, On the gravity side, consider a sub-interval α in AB, the RT surface of α is allowed to anchor on the cutoff brane (see the blue line), hence has two phases which are represented by the dashed and solid green lines.When α approaches AB, the solid green line has smaller length hence is the RT surface, and when we take the limit α → AB, the RT surface vanishes.This indicates that S AB = 0, hence AB is in a pure state.On the field theory side, one can apply the island formula and find that, when α → AB we have Is(α) → A ′ B ′ , which is consistent with the RT formula.
Secondly, from the island perspective the EWCS in W AB has a new interpretation.On the field theory side we should apply the island formula (1.1) to calculate S A , where A ′ = [q, a) and we have plugged in (4.3) in the third equation, which is the solution of the optimization.This solution of Is(A) coincides with (4.3), which the minimizes S AA ′ when calculating the EoP in [26].It was also pointed out in [2,41] that, the point x = q satisfying (4.3) is just the intersection point between the boundary and the geodesic (see the blue dashed line in Fig. 4) extended from the EWCS Σ AB .On the gravity side, the RT surface E A of A is just the geodesic emanating from the partition point x = p and anchored on the cutoff brane vertically (see the green line in the lower figure of Fig. 5), which coincide with the EWCS Σ AB in Fig. 4.This is consistent with the expectation that, when AB is in a pure state, the EWCS and its holographic dual quantites (like the EoP, the reflected entropy [42] and BPE), should coincide with S A .Thirdly, when we take AB is a sub-region of the interval, ρ AB is in a mixed state.We can also study the EWCS in W AB and its quantum information dual.From the island perspective, the cutoff brane plays the same role as the KR brane, then both of the RT surface E AB and the corresponding EWCS Σ AB undergo phase transitions since they are allowed to anchor on the cutoff brane.See Fig. 8 for example.On the field theory side, it was proposed in [41] that the so-called balance partial entanglement entropy (BPE), which is a partial entanglement entropy (PEE) satisfying certain balance conditions, corresponds to the EWCS on the gravity side.More interestingly, this correspondence was generalized to the holographic configurations with entanglement islands [23].We can calculate the BPE in island phases following the steps in [23], and check whether the BPE coincide with the EWCS.This is a highly non-trivial test for our proposal that the path-integral-optimized state |Ψ⟩ should be understood as in the island phase.Since the analysis of the BPE is complicated and relatively independent from the main topic of this paper, we put it in the appendix.Readers who are interested in the BPE and EWCS for the path-integraloptimized purification can go through the background papers [23,24,41] and the appendix in detail.
phase.This implication also extends to the path-integral optimized purification for an interval.The new perspective for the path-integral-optimized purification solves the puzzle of negative mutual information in this state.We calculated the entanglement entropies for subregions of the interval using the island formula, and find them coincide with the area of the RT surfaces that are allowed to anchor on the brane.Furthermore, we calculated the BPE between two arbitrary non-overlapping subregions of the interval in island phases, and find them coincide with the area of the EWCS.This gives non-trivial test for our proposal that the path-integral-optimized purification is a state in island phase.
Our results give a potential new link between the AdS/BCFT correspondence and the path integral optimization.This is quite important for the simulation of the AdS/BCFT configurations via the holographic Weyl transformed CFT, since the Weyl transformation is now determined on the field theory side, rather than adjusted by hand.Our calculations also provide new evidence for the correspondence between the BPE and the EWCS.

A The BPE/EWCS correspondence in the path-integral-optimized purification
In [2,41], it was proposed that the so-called balance partial entanglement entropy (BPE) is the quantum information quantity that duals to the EWCS.Furthermore, the concept of BPE was generalized to the island phases in [23] and its correspondence to the EWCS was also explicitly checked in the context of AdS/BCFT or the holographic Weyl transformed CFT 2 in [23,24].See also [52][53][54][55][56][57] for more discussion on the EWCS and its quantum information dual in holographic configurations with entanglement islands.In the appendix, we take the path-integral-optimized purification as a state in island phase, and study the entanglement wedges W AB for a bipartite subregion AB of the path-integral-optimized interval, with the cutoff brane playing the role of the KR brane in the context of AdS/BCFT.On the gravity side, we classify the phases of the EWCS of W AB and calculate the area of the EWCS in each phase.On the field theory side, we explicitly calculate the BPE between A and B. As expected, we find the agreement between the EWCS and the BPE in all the configurations.

A.1 Brief review for the PEE and BPE
For self-consistency, let us first introduce the basic concept of the partial entanglement entropy (PEE) [58][59][60].Some of the texts in this review overlap with section 3 in [24].The PEE is a measure of two-body correlation between two non-overlapping regions I(A, B).
Note that, we should not mix between the mutual information I(A, B) and the PEE I(A, B).So far, the fundamental definition for PEE based on the reduced density matrix is still not established.In some scenarios of interest, it can be determined by a set of physical requirements [60,61], which include all the properties satisfied by the mutual information and the additional key property of additivity: In the above list, A, B and C denote non-overlapping regions.For vacuum states of CFTs on a plane, we can determine the formula for PEE up to a coefficient by imposing the above requirements except the normalization property.Then we can determine the coefficient by imposing the normalization requirements for spherical regions where the relation between the geometric cutoff and the UV cutoff can be explored [62].
In this paper, we will use a particular construction for the PEE in generic twodimensional theories on a line or a circle 6 .This proposal is referred to as the additive linear combination (ALC) proposal [58,60,65], which claims that the PEE in these scenarios can be written as a linear combination of subset entanglement entropies that satisfy the property of additivity.
• The ALC proposal [58][59][60] : Consider a boundary interval A and partition it into three non-overlapping subregions A = α L ∪α∪α R , where α is some subregion inside A and α L (α R ) denotes the regions left (right) to it.On this configuration, the claim of the ALC proposal is the following: It is easy to check that, the above construction satisfies all the seven requirements in general [59,65].
The concept of PEE [58][59][60]65] originates from the study of the entanglement contour [63], which is defined as a function s A (x) that characterizes the contribution to the entanglement entropy of A from each site x ∈ A. By definition the entanglement contour function should satisfy, Figure 6: Here we consider the vacuum state of the holographic CFT2 on a circle which duals to the global AdS3, and consider two different partitions, where A and B are adjacent and disjoint respectively.
where σ x is a infinitesimal area element located at the site x.Subsequently, we can also define the contribution from a subset α in A to S A , The contribution s A (α) is a measure of the correlation between the subregion α and the complement Ā of A, which is exactly the two-body correlation we have defined as the PEE, Following [23], we call the notation on the left hand side of the above equation as the two-body correlation representation for the PEE, while the notation on the right hand side is the contribution representation.
The balanced partial entanglement entropy (BPE) is a special PEE that satisfies a set of balance requirements [41].In canonical purification, the definition of the BPE coincide with the reflected entropy [42], and the purification independence of the BPE was explored in [2].The BPE was proposed to be dual to the EWCS in holographic theories, and this proposal has undergone rigorous validation across diverse scenarios including both static [41] and covariant [68] setups of AdS 3 /CFT 2 with or without gravitational anomalies, and in the context of AdS/BCFT with entanglement islands [23,24].More interestingly, in the context of 3-dimensional flat holography [69][70][71], on the field theory side the BPE was calculated in [2,72], which matches the EWCS explored in [73].
For a system A ∪ B in a mixed state ρ AB , one can introduce an auxiliary system A ′ B ′ to purify AB such that the whole system ABA ′ B ′ is in a pure state |φ⟩ and, The pure state on ABA ′ B ′ is called a purification of ρ AB , which could be highly non-unique.
Let us consider the simple examples described in Fig. 6, where the balance requirements are summarized in the following [41]: Balance requirements: • When A and B are adjacent, the contribution from s AA 1 (A) and s BB 1 (B) should satisfy, This balance requirement is sufficient to determine the partition point in the purifying system A 1 B 1 , which we call the balance point.
• When A and B are non-adjacent, we have

and the balance requirements become
which are sufficient to determine the two partition points of the purifying system ) is automatically satisfied provided the satisfaction of the above requirements.
• Note that, it is possible that the solution to the balance requirements is non-unique.
In such cases, we should choose the solution that minimizes the BPE, which will be defined soon.
Provided the balanced requirements are satisfied, the BPE is then defined as Since we have I(A, B ′ ) = I(A ′ , B) at the balance point7 , the BPE can also be expressed as It is important to note that [2], the summation I(A, B ′ ) + I(A ′ , B) which we call the crossing PEE is minimized at the balanced point.This indicates that the BPE can be defined via an optimization problem.It is also interesting to note that, when A and B are adjacent, this minimal crossing PEE gives half of the lower bound of Markov gap [41,74,75], which is a universal constant (c/6) log 2. Is(B)

A.2 PEE and BPE in island phases
As described in the article [1], a system in the island phase can be understood as a selfencoding system.In other words, the state of certain subsets of the system are totally encoded in the state of their counterpart subsets of the system.Subsequently, the calculation of entanglement entropy should also be modified to the island formulas (1.1).Given the self-encoding property, when we compute the PEE between subregions in island phase, we should also take the contribution from the corresponding island regions into account.Now we generalize our construction of the PEE and BPE to configurations with entanglement islands [23].Here we just list the basic elements we need to carry out the computations.One should consult [23] for more details.Let us start with the region AB and the island regions Is(AB), Is(A) and Is(B).According to [23], the degrees of freedom in Is(AB) is not independent, and contributes to the entanglement entropy S AB calculated by the island formula.The PEE, for example s AB (A), should contain the contribution from the island region, and it is clear that we should assign the contribution from Is(A) to s AB (A) and similarly assign Is(B) to s AB (B).Nevertheless, there are scenarios with regions included in Is(AB) but outside Is(A) ∪ Is(B), which should also be assigned to s AB (A) or s AB (B).In other words, when Is(AB) ̸ = ∅ and Is(AB) ⊃ Is(A) ∪ Is(B), we define the ownerless island region to be Io(AB) = Is(AB)/(Is(A) ∪ Is(B)) . (A.11) The ownerless island should be further divided into two parts Io(AB) = Io(A) ∪ Io(B) , (A.12) which are assigned to s AB (A) and s AB (B) respectively.We will specify the division of Io(AB) later.All in all, we define the generalized islands and assign them to A and B respectively when calculating the PEEs.The generalized islands coincide with the so-called reflected entropy islands defined in [52].The assignment can be classified into three classes: • • Class 3: When we have non-trivial ownerless island region, then we have (A.13).
Now we turn to the computation of the PEEs for a bipartite system AB with entanglement island, and denote C ≡ AB ∪ Is(AB) for convenience.After taking into account the contributions from the islands, we should have [23], where the island contribution has been taken account.The right contribution comes from the generalized island rather than the entanglement island.A key step to compute the PEEs (A. 16) is to write the right hand side of (A.16) as a linear combination of the PEEs that can be written in this form I(γ, γ), i.e. a PEE between a region γ and its complement.For example, using the property of additivity, the PEE (A. 16) is just given by where we have used Ir(A)Ir(B) = Is(AB), and the notation Sγ ≡ I(γ, γ) which will be explained soon.
For the configurations shown Fig. 6, for example the adjacent case in the left figure, we should calculate s AA 1 (A) following (A.17).For the non-adjacent case in Fig. 6 where A is sandwiched by two regions A 1 and A 2 , we have We call this formula the generalized ALC formula [23] for island phases, which is just the ALC formula (A.1) with the replacement S γ ⇒ SγIr(γ) applied to each term.Accordingly, the balance requirements should also be modified to generalized versions, which are given by (A. 20) In summary, in island phases the BPE are still defined by (A.9), but we need to use the generalized versions of the ALC formula and balanced requirements.Before we compute the BPE, we need to clarify how to compute I(γ, γ).In this paper we need to deal with two types of γ, 1) Though the above two basic proposals have not been proved yet, they have produced highly consistent results between the BPE and the EWCS in various configurations with entanglement islands [23].The basic proposal 1 looks like a generalization of the RT formula in AdS 3 /CFT 2 for single interval.While the basic proposal 2 is a generalization of the RT formula for two-intervals with connected entanglement wedge.It looks reasonable as this proposal applies when the region [a, b] admits an island, hence the entanglement wedge looks more like the connected phase of a two-interval.Note that the basic proposal 2 is not consistent with the normalization and additive property of the PEE, which may be explained by a similar phase transition of the PEE flux in AdS 3 /CFT 29 [77,78].We leave this for future investigation [79].In summary, our calculations in this paper only involve linear combinations of Weyl transformed two-point functions (2.7) of the twist operators.The remaining problem is the division of the ownerless island region Io(AB) = Io(A) ∪ Io(B), which is indeed determined by the balance requirements.In [23], the authors considered the AdS/BCFT set-up and its simulation via a holographic Weyl transformed CFT 2 .They found that different assignments for the ownerless island region lead to different BPEs, which exactly correspond to different saddles of the EWCS.Then according to the minimum requirement, we should choose the assignment that gives us the minimal BPE.It seems that, the division of the ownerless island depends on the phase structure of the EWCS.

A.3 A case study for the path-integral-optimized purification
Here we give a typical example for the study of the EWCS and BPE in the path-integraloptimized purification of an interval, and assume that the cutoff brane plays the role of the KR brane and the purification is in island phase.We consider the path integral optimization for a single interval [a,b], hence the scalar field that characterizes the Weyl transformation is given by (3.7) with κ = 0, and the cutoff brane (3.9) is just a semi-circle.As a case study, we consider the setup shown in Fig. 8, where we considered two adjacent sub-intervals, whose size and position are adjusted such that, the RT surface E AB consists of two disconnected pieces of geodesics anchored on the cutoff brane, and the EWCS Σ AB also anchors on the brane.The lengths of the geodesics anchored on the cutoff brane are classified in appendix B. The cross-section of the entanglement wedge W AB has three saddle points, which anchors on either of the two disconnected pieces of the RT surface, or the cutoff brane, and the EWCS is the saddle point with minimal area.In the case shown in Fig. 8, the EWCS anchors on the cutoff brane.
Firstly, we check the self-consistency of the simulation by calculating the entanglement entropy on both sides of holography.In the gravity side, S AB is calculated by the RT formula, which is the area of the two pieces of geodesic anchored on the brane described as On the field theory side, we calculate S AB by island formula and denoting the island region requirement which settles q 1 to be Again, this point is also the intersection between the boundary and the extended EWCS Σ AB .Finally, we obtain the BPE for this phase as follows where we have plugged in (3.7) and (A.31) in the above equation.As expected, the BPE (A.32) precisely match with the area of the EWCS (A.27).Then we consider the scenarios where κ ̸ = 0 in (3.7), such that the cutoff brane is described by (3.9), which is no longer a semi-circle (see Fig. 9).The size, position and the partition of AB are adjusted such that, the RT surface E AB and the EWCS Σ AB all anchor on the cutoff brane.The lengths of the geodesics anchored on the cutoff brane are classified in appendix B. On the gravity side, the area of the EWCS can be easily calculated as follows Then we turn to the calculation for the BPE on the field theory side.Firstly we give an assignment for the island regions, Is(AB) = Ir(A) ∪ Ir(B), where where x = q 1 is the partition point of Is(AB), which will be determined by the balance requirements (A.19).Now we calculate the PEE on both sides of (A. 19), which have the same expression as (A.29) and (A.30), with the only difference that, the κ in the scalar field (3.7) is non-zero.Again the balanced condition is satisfied trivially and the implicit minimal requirement which settles q 1 to be Finally, we may obtain the BPE for this phase as follows where we have plugged in (3.7) and (A.35) in the above equation.As expected, the BPE expressed in (A.36) precisely match with the area of the EWCS given by (A.33).

B Geodesic Length from Geometrical Analysis
In this section, we give a generic way to derive the lengths for geodesic chords anchored on the boundary and terminated in the bulk, which we frequently encounter in this paper.

Let us consider two adjacent intervals
x 2 ] located in the vacuum CF T 2 which constitutes a pure state with the intervals , see Fig. 10.When the RT surface of A ′ A and AB are normal to each other, we can explicitly derive the lengths for the geodesic chords l 1 , l 2 , l ′ 1 and l ′ 2 in Fig. 10.To obtain the location of the point q ′ as indicated in Fig. 10, we consider a Euclidean triangle in the x-z plane and utilize trivial Euclidean identities.Consequently, the coordinate of the point q ′ are described as x = ρ 2 cos(θ ′ ) and z = ρ 2 sin(θ ′ ) or ρ 1 sin(θ) using the parameters in Fig. 11.We can also re-express the coordinates ρ 1 , ρ 2 and y in terms of the interval endpoints and applying cosine law in the triangle results in following relations, Euclidean triangle is considered in the x-z plane where ρ1 and ρ2 are the distances from the point q ′ and the center of the RT surfaces associated with the intervals A ∪ B and A ′ ∪ A respectively.The point y is the distance between the corresponding centers.
Note that, the two semi-circles depicted in Fig. 10 are normal to each other and therefore we can determine a constraint equation between the endpoint of the intervals x 1 , x 2 , x ′ 1 and x ′ 2 by utilizing right angle triangle identity ρ 2 1 + ρ 2 2 = y 2 as follows Furthermore, we establish a framework to compute the geodesic length l ′ 1 , then we further generalize this analysis to other geodesic segments.In this context, the geodesics equation for the segment l ′ 1 is described by , where C is the center for the semi-circle l ′ 1 + l ′ 2 .Now we utilize the geodesic length formula in pure AdS 3 geometry at constant time slice and integrate over the bulk coordinate z with limits involving endpoints of the geodesic segment l ′ 1 as Note that similar analysis can also be followed for the length of the geodesic segments l 1 , l 2 , and l ′ 2 .For each of the geodesic length segments shown in Fig. 10, the holographic proposal of the entanglement entropy discussed in [9] may be used to determine the holographic entanglement entropy from the (B.3) in a following way In the above equations, we have written the lengths for the geodesic chords in terms of the coordinates {x ′ 1 , x ′ 2 , x 1 , x 2 } of the boundary endpoints, which satisfy (B.2).In this paper, we utilize the expression of the corresponding geodesic segments to compute the lengths for the EWCS and other geodesic chords in various scenarios.

C The BPE/EWCS correspondence for adjacent AB with entanglement islands
Here we consider the configuration of two adjacent intervals where AB admits an island region.Again there are three possible assignments for the island regions: 1. A2a: 2. A2b: 3. A2c: In the phase-A2a, the interval AA 1 do not admit an island as shown in Fig. 12.The phase-A2c is symmetric to the phase-A2a.In the phase-A2b, both AA 1 and BB 1 admit an island, and the analysis is given in the main text.

Phase-A2a and phase-A2c
We now discuss the computation of the EWCS in the scenario depicted in Fig. 12.Here we have divided the complement of the adjacent intervals A ∪ B into A 1 ∪ B 1 with partition point p 1 .Utilizing the constraint (B.2) for the RT surfaces homologous to the intervals [q 1 , a 1 ] and [p 1 , p], we can obtain location of the point p 1 as In this scenario, the EWCS may be directly obtained from the length of the geodesic segment described in appendix B as follows Note that the value of q 1 can also be obtained in terms of the points a, a and b by using the constraint relation in (B.2) as Ir(B 1 ) Ir(B 1 ) Figure 12: Schematics shows the configuration of two adjacent intervals considered in Weyl CF T2 where the intervals B and B1 admit entanglement island region on the KR brane.Now we carry out the BPE analysis for this phase, where the island assignment Ir(B 1 ) = [q 1 , a] ∪ [b, q 2 ] and Ir(B) = (−∞, q 1 ] ∪ [q 2 , ∞) is shown in Fig. 12.In this phase, the balance requirement is given by I(A, BIr(B)B 1 Ir(B 1 )) =I(BIr(B), AA 1 ) . (C.4) We now utilize the generalized ALC proposal to compute PEEs mentioned in the above balance requirement, Solving the balance requirement (C.4) we get the partition point p 1 , which is same as (C.1).Finally, we can obtain the BPE using (C.5) and (C.1), in the phase can be obtained by utilizing (D.2) and (D.1) as follows .
(D.4) We now compute the BPE.As indicated in Fig. 13, where Ir(B 1 ) = (−∞, a] ∪ [b, ∞) occupies the whole island region.We need to determine the corresponding partition point via the balance requirement.In this context, the balance requirements are given by the following two equations (D.5) The above PEEs can be computed using the generalized ALC proposal as follows and From the balance requirement (D.5), we can compute the partition points p 1 and p 2 , which are the same as (D.4).The BPE for this phase-D1a may be obtained utilizing (D.4) and (D.6) as (D.10) Note that the above result of BPE can be exactly matched with the EWCS (D.3).

Phase-D1b
given by y 1 and y 2 as indicated in Fig. 14, where we used Euclidean triangle construction as discussed in appendix B for the RT surfaces homologous to the intervals [q 1 , p 2 ], [a 1 , b 2 ] and [q 1 , p 2 ], [a 2 , b 1 ] for calculating the y 2 and y 1 respectively.The minimal length at a constant time slice associated to these endpoints is given by (D.12) Utilizing (D.11) and (D.12), we may the compute the EWCS for this phase as follows where q 1 and p 2 are given by: (D.14) Note that q 1 can also be written in terms of the points a 2 , b 1 and p 2 using the constraint condition in (B.2) for the RT surfaces homologous to [q 1 , p 2 ] and [a 2 , b 1 ] as Ir(B 1 ) In this phase-D1b, the computation of BPE involves entanglement island regions Ir(A 1 ) = [q 1 , q 2 ] and Ir(B 1 ) = (−∞, q 1 ] ∪ [q 2 , a] ∪ [b, ∞) shown in Fig. 14.We now calculate the location of partition points p 1 and p 2 from balance requirements (D.16) The above PEEs may be computed using the generalized ALC proposal as follows and From the balance requirement (D.16), the partition points q 1 and p 2 are the same as (D.14).Finally, we may obtain BPE for this phase as follows The above result of BPE exactly matches with the EWCS in (D.13) using the value of partition point p 2 and q 1 given in (D.15) and (D.14).

D.2 Disjoint AB admits island
In this subsection, we consider the configuration of two disjoint intervals A = [a 1 , a 2 ] and B = [b 1 , b 2 ] where AB admits entanglement island region.We study the following assignments for the island region, The intervals A, A 1 and A 2 does not receive any contributions from the island region in the phase-D2a as shown in Fig. 15.The last phase-D2c involves all entanglement island regions for the all the intervals A, B, A 1 , B 1 , A 2 and B 2 .However in this phase, the entanglement wedge gets disconnected consequently the EWCS and BPE become zero.
We now compute the BPE in Fig. 15.Consequently, the balance requirements are given by I(A, BIr(B)B 1 Ir(B 1 )B 2 ) =I(BIr(B), A 1 A 2 A) , Utilizing the generalized ALC proposal, the above PEEs may be described as follows and Utilizing the balance requirements in (D.26), we get the solutions of p 1 and p 2 that are the same as (D.25).Finally we obtain the BPE for this phase-D2 satisfying the balance conditions as follows (D.32) The above result of BPE exactly coincide with the EWCS by putting the value of p 2 in (D.24).

Phase-D2b
As depicted in Fig. 16, the computation of the bulk EWCS involves endpoints y 2 and y 1 where y 2 is located on the KR brane.Here y 1 is situated on the RT surface homologous to the interval [a 2 , b 1 ].Note that the minimal length between the corresponding endpoints can be calculated using the following length formula in the AdS 3 geometry at a constant time slice as L = where the endpoints of the EWCS y 1 and y 2 are given by (D.34) Utilizing (D.34) and (D.33), we may compute the EWCS for this phase as follows (D. 35) where q 1 and p 2 are given by:  ]∪[q 1 ,q 3 ] + S[p 1 ,a 1 ]∪[q 2 ,q 3 ] − S[a 1 ,p 2 ]∪[q 1 q 2 ] = 1 2 S[q 1 ,p 2 ] + S[q 3 ,p 1 ] + S[q 2 ,a 1 ] + S[q 3 ,p 1 ] − S[q 2 ,a 1 ] − S[q 1 ,p 2 ] = c 6 log (p 1 − q 3 ) 2 As we can observe from the second balance constraint from above which are trivially satisfied.We may obtain q 1 from the first balance condition and it reduces in terms of the partition point p 2 .Therefore, we need to minimize the BPE for this phase over the partition point p 2 and it provides following value of q 1 and p 2 , which are the same as (D. (D.43) The above result of BPE exactly coincide with the EWCS in (D.35) by putting the value of q 1 .

Figure 1 :
Figure 1: Cut-off sphere at x = x0.For RT surfaces anchored at x0, the part inside the cutoff sphere is excluded, which has the same length |ϕ(x0)|

Figure 2 :
Figure 2: This figure shows the cutoff spheres and cutoff brane on a time slice for the Weyl transformed CFT with the scalar field (2.9).The radius of the cutoff sphere centered at x = x0 is given by α = −x0e −κ .

Figure 3 :
Figure3: This figure shows the numerical results of to series of cut-off spheres and the corresponding cutoff brane (dashed line).We have set a=1, b=2 and κ = 0, 1 in the left and right figures respectively.

Figure 4 :
Figure 4: The blue circles are the RT surfaces for AB and AA ′ .The Weyl transformation (3.7) with κ = 0 is performed on A ′ B ′ .When calculating S AA ′ , we should subject a constant term c 6 |ϕ(q)|, which is exactly the length of the dashed portion.Then S AA ′ exactly matches the length of the EWCS ΣAB.

Figure 5 :
Figure 5: Upper figure: RT surfaces for a sub-interval α in AB.Lower figure: RT surface for A.

Figure 7 :
Figure 7: Ownerless island in a time slice of holographic BCFT with one boundary.Here A2 ∪ B2 admits no island, and Is(AB) covers the whole brane.Nevertheless, the union of Is(A) and Is(B) does not cover the whole brane, which means there is an ownerless island Io(AB) inside Is(AB) but outside Is(A) ∪ Is(B).

Figure 10 :
Figure 10: Schematics shows the purification of the adjacent intervals A = [x1, x ′ 2 ] and B = [x ′ 2 , x2] located in the vacuum CF T2.The geodesics lengths homologous to the intervals A ∪ A ′ and A ∪ B are depicted as l ′ 1 + l ′ 2 (black) and l1 + l2 (blue) respectively and the intersection point of the corresponding geodesics is labelled as q ′ .

(D. 3 ) where p 1
and p 2 are given as follow through solving (B.2) for the RT surfaces homologous to [p 1 , p 2 ] and [a 1 , b 2 ], and the RT surfaces homologous to [p 1 , p 2 ] and [a 2 , b 1 ]